$1\mathrm{.}5$ A two$-$dimensional flow motion $($linear periodic gravity waves in water depth, $d)$ is defined in a Lagrangian system of coordinates by the equations:

$x=x_i+\frac{H}{2}\frac{coshm(d+z)}{sinhmd}sin(kt-mx)$

$z=z_i+\frac{H}{2}\frac{sinhm(d+z)}{sinhmd}cos(kt-mx)$

where $H$ is the wave height; $m,k,$ and $d$ are constants. $(m=2\frac{}{L}$;$L$ is the wavelength; $k=2\frac{}{T},T$ is the wave

period; and $d$ is the water depth.$)$ Find the equations for the streamlines, and sketch them. Assuming $|x-x_0|$ and $|z-z_0|$ are small, find the approximate equation for the particle paths. Sketch these paths.